Bjørn Gustavsen

Improving the pole relocating properties of vector fitting

This paper describes a modification of the vector fitting (VF) procedure for rational function approximation of frequency domain responses. The modification greatly improves the ability of VF to relocate poles to better positions, thereby improving its convergence performance and reducing the importance of the initial pole set specification. This is achieved by removing the high frequency asymptotic requirement of the scaling function used by VF. Calculated results demonstrate a major improvement of performance when fitting responses that are contaminated with noise. The procedure is also shown to be advantageous for transmission line modeling and wide band transformer modeling

Enforcing Passivity for Admittance Matrices Approximated by Rational Functions

A linear power system component can be included in a transient simulation as a terminal equivalent by approximating its admittance matrix Y by rational functions in the frequency domain. Physical behavior of the resulting model entails that it should absorb active power for any set of applied voltages, at any frequency. This requires the real part of Y to be positive definite (PD). We calculate a correction to the rational approximation of Y that enforces the PD criterion to be satisfied. The correction is minimal with respect to the fitting error. The method is based on linearization and constrained minimization by quadratic programming. Examples show that models not satisfying the PD criterion can lead to an unstable simulation, even though the rational approximation has stable poles only. Enforcement of the PD criterion is demonstrated to give a stable result.

A universal model for accurate calculation of electromagnetic transients on overhead lines and underground cables

This paper presents a transmission line model for the simulation of electromagnetic transients in power systems. The model can be applied to both overhead lines and cables, even in the presence of a strongly frequency dependent transformation matrix and widely different modal time delays. This has been achieved through a phase domain formulation where the modal characteristics have been utilized in the approximation for the propagation matrix. High computational efficiency is achieved by grouping modes with nearly equal velocities and by columnwise realization of the matrices for propagation and characteristic admittance.

Computer code for rational approximation of frequency dependent admittance matrices

This paper deals with the problem of approximating with rational functions a matrix whose frequency-dependent elements have been obtained from calculations or from measurements. Based on a previously developed technique (vector fitting), a set of callable routines have been written in the MATLAB language. These routines allow for rational approximation with a common set of stable poles, automatic selection of initial poles, passivity enforcement, and creation of an equivalent electrical network which can be imported into ATP-EMTP. Usage of sparse arithmetic permits the computer code to handle large systems. The methodology is demonstrated by application to a frequency-dependent network equivalent of a radial distribution network (phase domain), for which the accuracy is validated in both the frequency domain and the time domain. The computer code is in the public domain and is available from the author.

Wide band modeling of power transformers

This paper describes the measurement setup and modeling technique used for obtaining a linear wide band frequency-dependent black box model of a two-winding power transformer, for the purpose of calculation of electromagnetic transients in power systems. The measurement setup is based on a network analyzer, shielded cables, and a connection board. The setup is demonstrated to give a consistent data set where the effect of the measurement cables can be eliminated. The accuracy of the data set is increased by using a combination of current measurements and voltage transfer measurements. A rational approximation of the admittance matrix is calculated in the frequency domain in the range of 50 Hz to 1 MHz and subjected to passivity enforcement, giving a stable model which can be included in electromagnetic transients program (EMTP)-type simulation programs. The accuracy is validated both in the frequency domain and in the time domain.

Fast Passivity Enforcement for Pole-Residue Models by Perturbation of Residue Matrix Eigenvalues

Rational models must be passive in order to avoid unstable time domain simulations. This paper introduces a fast approach for passivity enforcement of pole-residue models. This is achieved by perturbing the eigenvalues of the residue matrices, as opposed to the existing approach of perturbing matrix elements. This leads to large savings in computation time with only a small increase of the modeling error. This fast residue perturbation (FRP) approach is merged with the modal perturbation technique, leading to fast modal perturbation (FMP). Usage of FMP over FRP achieves to retain the relative accuracy of the admittance matrix eigenvalues. A complete approach is obtained by combining the passivity enforcement step with passivity assessment via the Hamiltonian matrix eigenvalues and a robust iteration scheme, giving a guaranteed passive model. Application of FMP to a six-port power transformer shows that the approach is able to remove large out-of band passivity violations without corrupting the in-band behavior. This is shown to mitigate an unstable simulation. The approach is also demonstrated for a high-speed interconnect and a transmission line.

A Half-Size Singularity Test Matrix for Fast and Reliable Passivity Assessment of Rational Models

One major difficulty in the rational modeling of linear systems is that the obtained model can be nonpassive, thereby leading to unstable simulations. The models passivity properties are usually assessed by computing the eigenvalues of a Hamiltonian matrix, which is derived from the model parameters. The purely imaginary eigenvalues represent crossover frequencies where the models conductance matrix is singular, allowing to pinpoint frequency intervals of passivity violations. Unfortunately, the eigenvalue computation time can be excessive for large models. Also, the test applies only to symmetrical models, and the testing is made difficult by numerical noise in the extracted eigenvalues. In this paper a new (non-Hamiltonian) half-size singularity test matrix is derived for use with admittance parameter state-space models, which overcomes these shortcomings. It gives a computational speedup by a factor of eight; it is applicable to both symmetric and unsymmetrical models; and it produces noiseless eigenvalues for reliable passivity assessment.

Advancements in Iterative Methods for Rational Approximation in the Frequency Domain

Rational approximation of frequency-domain responses is commonly used in electromagnetic transients programs for frequency-dependent modeling of transmission lines and to some extent, network equivalents (FDNEs) and transformers. This paper analyses one of the techniques [vector fitting (VF)] within a general iterative least-squares scheme that also explains the relation with the polynomial-based Sanathanan-Koerner iteration. Two recent enhancements of the original VF formulation are described: orthonormal vector fitting (OVF) which uses orthonormal functions as basis functions instead of partial fractions, and relaxed vector fitting (RVF), which uses a relaxed least-squares normalization for the pole identification step. These approaches have been combined into a single approach: relaxed orthonormal vector fitting (ROVF). The application to FDNE identification shows that ROVF offers more robustness and better convergence than the original VF formulation. Alternative formulations using explicit weighting and total least squares are also explored.

Dynamic System Equivalents: A Survey of Available Techniques

This paper presents a brief review of techniques available for reducing large systems to smaller equivalents. The paper is divided into High Frequency Equivalents, Low Frequency Equivalents, and Wide-band Equivalents.

A robust approach for system identification in the frequency domain

Accurate modeling of power system components for the purpose of electromagnetic transient calculations requires the frequency dependence of components to be taken into account. In the case of linear components, this can be achieved by identification of a terminal equivalent based on rational functions. This paper addresses the problem of approximating a frequency dependent matrix H(s) with rational functions for the purpose of obtaining a realization in the form of matrices A, B, C, D as used in state equations. It is shown that usage of the Vector Fitting approach leads to a realization in the form of a sum of partial fractions with a residue matrix for each pole. This can be directly converted into a realization in the form A, B, C, D in which B is sparse and each pole is repeated n times with n by n being the size of H. The number of repetitions can be strongly reduced and sometimes completely avoided by reducing the rank of the residue matrices, thereby producing a compacted realization which is physically more correct and also permits faster time-domain simulations. The error resulting from the rank-reduction can be reduced by subjecting the realization to a nonlinear least-squares procedure, e.g., Gauss-Newton as was used in this work.